Data sufficiency is a question type you’ve never seen before. This chapter will show you how to use basic POE techniques to make this format your new favorite kind of math.
Almost half of the thirty-seven math questions on the GMAT will be data-sufficiency questions. We’re about to show you how to use POE to make this strange question type easy.
If you’ve never heard of data sufficiency, that’s because these questions exist on no other test in the world and they definitely require some getting used to. If you have already taken a GMAT practice exam, or the actual GMAT, you may have spent several minutes just trying to understand the directions for data-sufficiency questions.
However, data-sufficiency questions really just test the same math concepts as problem-solving questions, but with a twist—a strange question format.
Here’s what a data-sufficiency question will look like on the GMAT:
What is the value of y ?
(1) y is an even integer such that −1.5 < y < 1.5
(2) Integer y is not prime.
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Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. |
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Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. |
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BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. |
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EACH statement ALONE is sufficient. |
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Statements (1) and (2) TOGETHER are NOT sufficient. |
Every data-sufficiency question consists of a question followed by two statements. There are also five possible answer choices, as shown. The answers are the same for every data-sufficiency question, so once you learn what each means, you won’t need to spend time rereading the answers. You’ll just be able to think about them as answers A, B, C, D, and E, which is how we’ll refer to them.
Notice that there are two words that the answer choices keep repeating—alone and sufficient. So, it looks like we’re supposed to evaluate the statements on their own—at least at first. Moreover, our task is evidently to determine whether we have sufficient information to answer the question.
That’s how data sufficiency differs from problem solving. In problem-solving questions you are asked to give a numerical answer to the question. In fact, the inclusion of five numerical answer choices tells you that you can assume that the question can be solved. For data-sufficiency questions, however, you’re not being asked to solve the question but to decide WHETHER the question can be solved. It may, in fact, turn out that the statements do not provide sufficient information to answer the question.
The first answer choice—which we’ll call answer A—indicates that we should first look at Statement (1) by itself to see if it is sufficient to answer the question.
In fact, the best way to work data-sufficiency problems is to look at one statement at a time. So, ignore Statement (2). Here, we’ve replaced Statement (2) with question marks to indicate that we are looking only at the first statement—almost as though we had covered up the second statement.
What is the value of y ?
(1) y is an even integer such that −1.5 < y < 1.5
(2) ????
Now, we’re ready to evaluate Statement (1) alone. There are three integers between −1.5 and 1.5: −1, 0, and 1. Of those, as you may recall from Chapter 7, only 0 is even. So, Statement (1) does provide sufficient information to answer the question.
We’re not ready to choose the first answer—answer A—yet, however, because the second part of the answer choice states that Statement (2) alone is not sufficient. Now, forget that you have ever seen Statement (1).
What is the value of y ?
(1) ????
(2) Integer y is not prime.
The second statement only tells us that y is not prime. So, possible values for y include 1, 4, 6, 8, etc. Do we know the value of y ? No way. So, Statement (2) is not sufficient. Because (1) is sufficient and (2) is not, the answer to this question is
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Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. |
Or, in other words, the correct answer is A.
Now that you’ve seen and worked a data-sufficiency question, it’s time learn how to make this weird question type your own. The first step is to understand what each of the answer choices means.
By making small changes to the example you’ve just seen, we can provide examples of each of the answer choices. Next to each example, you’ll find a graphic that provides a quick, down and dirty way to understand and remember each answer choice. Here’s the example for choice A again:
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Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. |
What is the value of y ?
(1) y is an even integer such that
−1.5 < y < 1.5
(2) Integer y is not prime.
Now, let’s make some changes to the statements, to get an example of answer B.
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Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. |
What is the value of y ?
(1) Integer y is not prime.
(2) y is an even integer such that
−1.5 < y < 1.5
As you can see from this example, choice B is pretty much the flip side of choice A. In this case, the first statement provides no help in determining the value of y, but the second statement tells us that y = 0.
A few more changes produce an example of answer C.
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BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. |
What is the value of y ?
(1) y is an even integer.
(2) −1.5 < y < 1.5
The first statement tells us that y is even, but there are a lot of even integers. The second statement gives us a range of values for y, but, by itself, we don’t even know that y is an integer from the second statement. So, neither statement is sufficient on its own. But, when we put them together, we know that y = 0.
Now, let’s get an example of answer D.
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EACH statement ALONE is sufficient. |
What is the value of y ?
(1) y is an even integer such that
−1.5 < y < 1.5
(2) For any integer a ≠ 0, ay = 0
As pointed out in previous examples, the information in Statement (1) allows us to conclude that y = 0. The information in the second statement also tells us that y is 0 because the only way for the product of ay to equal 0 is if either a or y is 0. Since a can’t be 0, y must be 0. Note how the statements independently allow us to arrive at the conclusion that y = 0 for answer choice D.
Finally, let’s look at an example of choice E.
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Statements (1) and (2) TOGETHER are NOT sufficient. |
What is the value of y ?
(1) y is an even integer.
(2) Integer y is not prime.
For this example, there’s no way to determine the value of y. The first statement doesn’t work because y could be any even integer. The second statement also doesn’t help because y can be any integer that isn’t prime. Even when we combine the statements, we don’t know the value of y because any even integer except 2 fits the conditions. So, E is the no way, no how answer.
Below, you’ll find the full graphic for all of the answers. You may find it helpful to keep the graphic handy until you are completely comfortable with what each answer choice means.
One of the reasons the test writers decided to include data-sufficiency questions on the GMAT is that when this format was first dreamed up they thought these questions would be immune to Process of Elimination (POE). Were they ever wrong! If anything, it’s even easier to apply POE to data-sufficiency questions. Let’s see why.
First, however, let’s restate one of the most important strategies for working any data-sufficiency question: Evaluate the statements one at a time before you think about combining them. Many people mistakenly pick C—you need both statements together—when it would have been possible to answer the question with only the information in the first statement or the second statement. Generally, people make this mistake when they read both statements right after reading the question stem. In fact, this mistake is the most common mistake that test takers make when working data-sufficiency questions.
To avoid this common mistake, read the question stem and only the first statement. Ignore the second statement. Pretend it isn’t there. You may even go as far as covering Statement (2) with your finger if you find the temptation to read both statements too overpowering. Once you have evaluated Statement (1), forget it. Ignore it. It doesn’t exist anymore. Cover it up if you need to and read and evaluate Statement (2).
What happens when you evaluate the statements one at a time? Something magical, that’s what! POE comes roaring back. Consider the following partial example:
What is the value of x ?
(1) x + 7 = 12
We don’t even have Statement (2), but we can still do a lot with this partial question. (Don’t worry. There won’t be any partial questions on the real test!) First, you want to see if the statement is sufficient to answer the question. In this case, you could subtract 7 from both sides of the equation to discover that x = 5. We’ll take this as an opportunity to remind you, however, that you don’t really need to solve the equation—you just need to know that you can solve the equation. After all, to pick an answer to the problem, you just need to know if you have sufficient information.
Since Statement (1) is sufficient in this case, which answer choices can be eliminated? From the chart, you can see that there are only two answer choices—A and D—that have Statement (1) circled to indicate that, for that answer choice, Statement (1) is sufficient. So, if the first statement is sufficient, the answer to the problem must be A or D!
What is the value of x ?
(1) x is an integer.
Now, what are the possible answers? If you said B, C, or E, you are well on your way to getting this data-sufficiency stuff under control. If you said something else, take a look at the chart back on this page. In this case, the first statement is insufficient to determine the value of x. So, you want the answer choices that have 1 crossed off, and that is B, C, or E.
In the following drill, each question is followed by only one statement. Based on the first statement, decide if you are down to AD or BCE. The answers can be found in Part V.
| 1. | What is the value of x ? |
(1) y = 4
(2) ????
| 2. | Is y an integer? |
(1) 2y is an integer.
(2) ????
| 3. | A certain room contains 12 children. How many more boys than girls are there? |
(1) There are three girls in the room.
(2) ????
| 4. | What number is x percent of 20 ? |
(1) 10 percent of x is 5.
(2) ????
Every time you start a data-sufficiency question, you should read the question and only the first statement. If the first statement is sufficient, your possible answers are A or D. If the first statement is insufficient, your possible answers are B, C, or E. So, you can always get rid or either two or three answer choices just by evaluating the first statement. The AD/BCE split is so important that you’ll want to write down AD or BCE on your noteboard.
But what happens next? How do you get to the answer? Let’s take a look.
If x + y = 3, what is the value of xy ?
(1) x and y are integers
(2) x and y are positive
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Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. |
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Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. |
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BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient. |
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EACH statement ALONE is sufficient. |
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Statements (1) and (2) TOGETHER are NOT sufficient. |
As always, ignore Statement (2) and look only at Statement (1). If x and y are integers and x + y = 3, do we know what they are? Not really—x could be 1 and y could be 2 (in which case, xy would be 2). But x could also be 0 (yes, 0 is an integer) and y could be 3 (in which case, xy would be 0). Because Statement (1) alone does not answer the question definitively, we are down to BCE, a one in three shot. Write it down in your scratch booklet.
Now, ignore Statement (1) and look at Statement (2). By itself, this statement doesn’t begin to give us values for x and y—x could be 1 and y could be 2, but x could just as easily be 1.4 and y could be 1.6. Because there is still more than one possible value for xy, cross off answer choice B.
We’re down to C or E. Now it’s finally time to look at both statements at the same time. See how late in the process we combine the statements? Get into the habit of physically crossing off B before you think about combining the statements. That’s how you can avoid making the most common GMAT data-sufficiency mistake of putting the statements together too early.
Because we know from the first statement that x and y are integers, and from the second statement that they must be positive, do we now know specific values for x and y ?
Well, we do know that there are only two positive integers in the world that add up to 3: 2 and 1. (Remember, zero is an integer but it is neither positive nor negative.)
Do we know if x = 1 and y = 2, or vice versa? Not really, but frankly, it doesn’t matter in this case. The question is asking us the value of xy.
Because neither statement by itself is sufficient, but both statements together are sufficient, the answer is choice C.
Here’s a handy flowchart that shows you what to do for any data-sufficiency problem. You should keep the flowchart next to you and consult it as you first start practicing data-sufficiency questions. After you have done ten or twenty questions, you’ll probably find that you have learned the basic POE process well enough that you don’t need the chart anymore. However, if you ever find yourself having trouble with data sufficiency, pull out the chart again and do some more problems using it as a guide.
If you were going to provide the answer to most data-sufficiency questions, your response would be a number. However, as many as half of all the data-sufficiency questions that you will see on your test will ask a yes-or-no question instead.
Leave it to GMAC to come up with a way to give you five different answer choices on a yes-or-no question. Let’s look at an example.
Did candidate x receive more than half of the 30,000 votes cast in the general election?
(1) Candidate y received 12,000 of the votes cast.
(2) Candidate x received 18,000 of the votes cast.
When all is said and done, the answer to this question is either yes or no. Start by ignoring Statement (2) and evaluating Statement (1). Does Statement (1) alone answer the question? If you were in a hurry, you might think so. Many people assume that there are only two candidates in the election. They reason that if candidate y got 12,000 votes, then candidate x must have received 18,000 votes. However, there’s no reason to assume that there are only two candidates. So, Statement (1) is insufficient. Write down BCE. Does Statement (2) alone answer the question? Yes, it’s pretty clear that candidate x received more than half of the votes. So, the correct answer is B.
That didn’t seem so bad, did it? Yet, you may have heard that yes/no data-sufficiency questions have a reputation for being hard. Let’s change our example to see why.
Did candidate x receive more than half of the 30,000 votes cast in the general election?
(1) Candidate y received 12,000 of the votes cast.
(2) Candidate x received 13,000 of the votes cast.
As always, start by ignoring Statement (2) so that you can properly evaluate Statement (1) alone. As in our previous example, Statement (1) is insufficient, so be sure to write down BCE. Statement (2) seems pretty straightforward. Candidate x received fewer than half of the votes cast. At this point many people say, “Since the guy clearly got fewer than half the votes, this statement doesn’t answer the question, either.” But those people are wrong!
Broken down to its basics, the question we were asked was, “Did he get more than half of the vote—yes or no?”
Statement (2) does answer the question. The answer is, “No, he didn’t.” So, the answer is the same as that of the first example. The answer is B.
On a yes/no data-sufficiency problem, if the statement answers the question in either the affirmative or the negative, it is sufficient.
Yes/no questions really should be called yes/no/maybe questions. Even if that’s not their “official” name, it’s still worthwhile to think about them in that fashion.
Let’s look at one last example to see why.
Did candidate x receive more than half of the 30,000 votes cast in the general election?
(1) Candidate y received 12,000 of the votes cast.
(2) Candidate x received at least 13,000 of the votes cast.
Since the first statement of this question is the same as that of the previous two examples, we know that it is insufficient. So, write down BCE. Now, let’s tackle Statement (2). Based on Statement (2), candidate x could have received exactly 13,000 votes, which would make the answer to the question “No, he did not receive more than half the votes cast.” However, he could have also received 16,000 votes, and that would make the answer to the question, “Yes, he did receive more than half the votes cast.” So, based on Statement (2), the best we can really say is that candidate x may have received more than half the votes. “Maybe” isn’t good enough—we need a definitive yes or no answer. So, Statement (2) is insufficient. Cross off B. What if we combine the statements? We still have the same problem. We’ve accounted for at least 25,000 of the votes between the two candidates, but we don’t know about the other 5,000. All of those votes could have gone to candidate x, making the answer to the question “yes.” However, there could have been a third candidate who received those 5,000 votes. In that case, x would have received only 13,000 votes and the answer to the question is “no.” Combining the statements didn’t get us to a definitive answer. Cross off C. The correct answer is E.
Although data sufficiency problems test the same material covered by regular problem-solving questions, some readers find it distracting to learn the more complicated subtleties of this new question type at the same time that they are learning (or relearning) math concepts. That’s why we’ve put our main chapter on data sufficiency at the end of our math review.
However, you will find data-sufficiency problems sprinkled throughout the math drills—and you should feel free at any time to dip into Chapter 14, where you’ll find everything in one place, including more advanced strategy, several more drills, and some great techniques to handle the most complicated yes/no questions.